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We introduce a data driven and model free approach for computing conditional expectations. The new method combines Gaussian Mean Mixture models with classic analytic techniques based on the properties of the Gaussian distribution. We also incorporate a proxy hedge that leads to analytic expressions for the hedge with respect to the chosen proxy. This essentially makes use of the representation of the hedge sensitivity measuring the part of the variance that is attributed to the proxy. If we take the underlying, this corresponds to a time discrete minimal variance delta hedge. We apply our method to the calibration of pricing and hedging of (multi-dimensional) exotic Bermudan options, the calibration of stochastic local volatility models and applications to xVA/exposure calculation. For illustration we have chosen the rough Bergomi model and high-dimensional Heston models. Finally, we discuss issues when increasing the dimensionality and propose solutions using established statistical learning methods.